We have a tendency to evaluate all anxiety as being sinful. And I would generally agree with that evaluation. However, there is some anxiety that is Biblically appropriate because it focuses on God and His people. Let me explain how the Bible displays this type of anxiety in a good light.
First, remember Paul’s words to the Galatian Christians in 4:20. He writes, “I am perplexed [aporeo Strong's # 639] about you.” We are reminded of the word picture indicating that we have reached the bank of the river and are unable to find the fording place. We are at a loss as to what to do next. We don’t know how to proceed in this situation. This uncertainty unsettles us, perplexes and disquiets us. There is no hint of sin at this stage. In fact, this sort of occasion can be healthy for us because it encourages us to turn to the only wise God for guidance. (cf. Proverbs 3:5-6; Romans 16:27; 1st Timothy 1:17; Jude 1:25) You may want to look at the other uses of this verb (cf. John 13:22; Acts 25:20; and 2nd Corinthians 4:8) and the noun [aporia # 640] (cf. Luke 21:25).
Second, Paul reminds the Corinthian believers of his concern for all the congregations he has worked with (2nd Corinthians 11:28). Here Paul utilizes a Greek word for concern [merimna # 3308]. You’ll recognize from the context that Paul is detailing indicators of his loving concern. Look at what he experienced while ministering to them. This is not sinful apprehensiveness, but appropriate concern for those we love. Sinful worry tends to focus on future possibilities of danger or misfortune. Legitimate concern knows that potential dangers exist and is concerned for those loved ones who may experience these potential dangers. Appropriate concern is referred to in 1st Corinthians 7:32-34. Sinful worry is condemned in Matthew 6:25ff and Philippians 4:6.
So, can pastors have legitimate anxiety for the flock among which they minister? Yes. Thank God that your pastor genuinely cares about the flock. Pray for your pastor that this anxiety will not shift over into the realm of sinful worry. Legitimate anxiety turns in faith to the omnipotent God, who works all things after the counsel of His will (cf. Ephesians 1:11). Sinful worry turns to the impotent resources of self.